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Theorem eldprdi 19621
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0 0 = (0g𝐺)
eldprdi.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
eldprdi.1 (𝜑𝐺dom DProd 𝑆)
eldprdi.2 (𝜑 → dom 𝑆 = 𝐼)
eldprdi.3 (𝜑𝐹𝑊)
Assertion
Ref Expression
eldprdi (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
Distinct variable groups:   ,𝐹   ,𝑖,𝐺   ,𝐼,𝑖   0 ,   𝑆,,𝑖
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝑊(,𝑖)   0 (𝑖)

Proof of Theorem eldprdi
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . 2 (𝜑𝐺dom DProd 𝑆)
2 eldprdi.3 . . 3 (𝜑𝐹𝑊)
3 eqid 2738 . . 3 (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)
4 oveq2 7283 . . . 4 (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹))
54rspceeqv 3575 . . 3 ((𝐹𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
62, 3, 5sylancl 586 . 2 (𝜑 → ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
7 eldprdi.2 . . 3 (𝜑 → dom 𝑆 = 𝐼)
8 eldprdi.0 . . . 4 0 = (0g𝐺)
9 eldprdi.w . . . 4 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
108, 9eldprd 19607 . . 3 (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
117, 10syl 17 . 2 (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
121, 6, 11mpbir2and 710 1 (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  {crab 3068   class class class wbr 5074  dom cdm 5589  cfv 6433  (class class class)co 7275  Xcixp 8685   finSupp cfsupp 9128  0gc0g 17150   Σg cgsu 17151   DProd cdprd 19596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-ixp 8686  df-dprd 19598
This theorem is referenced by:  dprdfsub  19624  dprdf11  19626  dprdsubg  19627  dprdub  19628  dpjidcl  19661
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